# The minimizing of the Nielsen root classes

Daciberg Gonçalves; Claudemir Aniz

Open Mathematics (2004)

- Volume: 2, Issue: 1, page 112-122
- ISSN: 2391-5455

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topDaciberg Gonçalves, and Claudemir Aniz. "The minimizing of the Nielsen root classes." Open Mathematics 2.1 (2004): 112-122. <http://eudml.org/doc/268702>.

@article{DacibergGonçalves2004,

abstract = {Given a map f: X→Y and a Nielsen root class, there is a number associated to this root class, which is the minimal number of points among all root classes which are H-related to the given one for all homotopies H of the map f. We show that for maps between closed surfaces it is possible to deform f such that all the Nielsen root classes have cardinality equal to the minimal number if and only if either N R[f]≤1, or N R[f]>1 and f satisfies the Wecken property. Here N R[f] denotes the Nielsen root number. The condition “f satisfies the Wecken property is known to be equivalent to |deg(f)|≤N R[f]/(1−χ(M 2)−χ(M 10/(1−χ(M 2)) for maps between closed orientable surfaces. In the case of nonorientable surfaces the condition is A(f)≤N R[f]/(1−χ(M 2)−χ(M 2)/(1−χ(M 2)). Also we construct, for each integer n≥3, an example of a map f: K n→N from an n-dimensionally connected complex of dimension n to an n-dimensional manifold such that we cannot deform f in a way that all the Nielsen root classes reach the minimal number of points at the same time.},

author = {Daciberg Gonçalves, Claudemir Aniz},

journal = {Open Mathematics},

keywords = {55M20; 57M12; 20F99},

language = {eng},

number = {1},

pages = {112-122},

title = {The minimizing of the Nielsen root classes},

url = {http://eudml.org/doc/268702},

volume = {2},

year = {2004},

}

TY - JOUR

AU - Daciberg Gonçalves

AU - Claudemir Aniz

TI - The minimizing of the Nielsen root classes

JO - Open Mathematics

PY - 2004

VL - 2

IS - 1

SP - 112

EP - 122

AB - Given a map f: X→Y and a Nielsen root class, there is a number associated to this root class, which is the minimal number of points among all root classes which are H-related to the given one for all homotopies H of the map f. We show that for maps between closed surfaces it is possible to deform f such that all the Nielsen root classes have cardinality equal to the minimal number if and only if either N R[f]≤1, or N R[f]>1 and f satisfies the Wecken property. Here N R[f] denotes the Nielsen root number. The condition “f satisfies the Wecken property is known to be equivalent to |deg(f)|≤N R[f]/(1−χ(M 2)−χ(M 10/(1−χ(M 2)) for maps between closed orientable surfaces. In the case of nonorientable surfaces the condition is A(f)≤N R[f]/(1−χ(M 2)−χ(M 2)/(1−χ(M 2)). Also we construct, for each integer n≥3, an example of a map f: K n→N from an n-dimensionally connected complex of dimension n to an n-dimensional manifold such that we cannot deform f in a way that all the Nielsen root classes reach the minimal number of points at the same time.

LA - eng

KW - 55M20; 57M12; 20F99

UR - http://eudml.org/doc/268702

ER -

## References

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